A key falls from a bridge that is 45 m above the water. It falls directly into a model boat, moving with constant velocity, that is 12 m from the point of impact when the key is released . What is the speed of the boat ?

Answer

To solve the problem of the key falling from a bridge and landing in a moving boat, we can break down the solution into clear steps:

### Step-by-Step Solution:

1. **Identify the Given Data:**
- Height of the bridge (h) = 45 m
- Distance of the boat from the point of impact (d) = 12 m
- Acceleration due to gravity (g) = 10 m/s² (assuming downward direction)

2. **Determine the Time of Fall:**
The key falls from a height of 45 m. We can use the second equation of motion to find the time (t) it takes for the key to fall:
\[ s = ut + \frac{1}{2} a t^2 \]
Here,
- \(s = -45\) m (downward displacement, hence negative),
- \(u = 0\) m/s (initial velocity),
- \(a = -g = -10\) m/s² (acceleration due to gravity).

Plugging in the values:
\[ -45 = 0 \cdot t + \frac{1}{2} \cdot (-10) \cdot t^2 \]
Simplifying:
\[ -45 = -5t^2 \]
\[ t^2 = 9 \quad \Rightarrow \quad t = 3 \text{ seconds} \]

3. **Calculate the Speed of the Boat:**
The boat needs to cover a distance of 12 m in the same time it takes for the key to fall, which is 3 seconds. We can use the formula for speed:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]
Substituting the known values:
\[ \text{Speed} = \frac{12 \text{ m}}{3 \text{ s}} = 4 \text{ m/s} \]

4. **Conclusion:**
The speed of the boat must be 4 m/s to ensure that it reaches the point directly under the key when it falls.

### Final Answer:
The speed of the boat is **4 m/s**.